Using multi-beam optical tweezers to create optical lattices and manipulate micro-particles
Introduction
Light can transfer linear and angular momenta to objects in its way, resulting in exerting observable force and torque on them [1]. The first utilization of optical force was demonstrated by Ashkin in 1970 [2], who accelerated and trapped micro-particles by employing two counter laser beams. Later on, in 1986, he and coworkers reported the optical trapping of a dielectric particle by a single highly focused laser beam [3], known as an optical tweezer [4]. In optical tweezers, a single laser beam is highly focused using a high numerical aperture objective lens to produce an axial intensity gradient. The optical manipulation of micro-particles has become effective tools broadly used in various fields of science, specifically in physics [5], chemistry [6], biology [7], colloidal science [8], and engineering [9]. Furthermore, it has been used in a wide range of applications, including optical trapping [10], optical sorting [11], creation of optical lattice [12], stretching [13], acceleration [14] and transportation [15], etc. The high resolution and contactless force are advantages of the optical manipulation technique over other manipulation techniques [16]. Nowadays, the target of optical trapping is not limited to the single-particle. One can achieve a wide variety of methods and setup for different types of multi-particle trapping for various applications like manipulating numbers of the particle [17], creating particle arrays [18], optically sorting particles [19], and so on. Multiple traps can be achieved in different ways, such as holographic optical tweezers [20] or scanning optical tweezers [21]. Further, to produce multi-particle manipulation, one can employ optical lattices. Optical lattices are composed of interfering beams and produce periodic arrays of trapping sites [22]. Optical lattices created by interfering with two or three beams were reported [23]. Mohammanezhad et al. simulated the effects of the different number of beams and their polarization on a Rayleigh particle in the optical lattice resulting from multi-beam interference [24]. Furthermore, Ha and coworkers studied how polarization affects optical force on a spherical particle with arbitrary size in a three-beam optical lattice [25]. In this paper, we study the optical force acting on a spherical particle with arbitrary size (Mie particle, which is commonly used in experiments and Rayleigh particle) positioned in N beam optical lattice formed by interference of N identical traveling plane waves incident directly on the boundary of two semi-infinite dielectric media. The configuration that is used to achieve the desired optical lattice is shown in Fig. 1.
The configuration shown in Fig. 1 can be used to create the optical lattice with diverse intensity patterns over the interface of two dielectric media. Particles on the interface are affected by this intensity pattern hence force is applied to them. We focus on how the number of beams and their associated polarization and their incident angle affect the force landscape profile. Also, the potential energy of particles in this interference pattern is calculated. The decomposition of optical force is extremely important to optical tweezer, and it will help us understand the physical mechanism during the multi-beam trapping. And also, calculating the gradient and scattering parts of the optical force is momentous because they are two constituents in optical force with broad applications. The gradient force can be used for trapping, and scattering force can be employed as propulsion [26], pulling [27], light-driven motors [28], and so on. In two limited regimes where the particle is either much larger(ray-optic approximation) [29] or much smaller(dipole approximation) [30] than the operating light wavelength, approximation models are suitable and easily carried for partitioning the optical force into the gradient and scattering parts. In the intermediate regime, the decomposing optical force acting on Mie particles, which are the most accessible in the optical manipulation, remains open. In this regime, the mechanical effects of light can be studied with the generalized Lorenz-Mie theory (GLMT) [31,32]. Unfortunately, Decomposition of the force into the gradient and the scattering part for the Mie particles by GLMT has not yet been reported. Nevertheless, for Rayleigh particles, this has been done [33,34]. Many approaches have been made to decompose gradient force and scattering force from the total force [35,36]. Still, these approaches have problems such as large amounts of computation, or the dependence of two force components on the illuminating optical fields is unclear in analytical expressions [37]. Based on the Cartesian multipole expansion theory in [37], [38], [39], [40], we investigate the gradient and scattering parts of total force acting on spherical particles with the arbitrary radius from multi beams optical lattice. In the interference approach, by increasing the number of incident beams, one can attain a tiny spot intensity beyond the diffraction limit without using a high numerical aperture objective lens, and this is an advantage of this approach [22]. The multi beams interference setup may find many applications in creating optical matter or optical arrangement of small particles or controlling colloidal crystals evolution [41], [42], [43]. The results of our research, as well as what is in the [24,44], can be a good guide for experimental investigations in the field of multi-beam interference trapping.
Section snippets
Theory
N collimated plane-wave beams move directly to the common point in the boundary surface of two media (origin of our system) with incident angle θ, and the angle between each adjacent beam is , the polystyrene particle (np = 1.59) is locating on the glass (n2 = 1.51) and immersed in air (n1 = 1) in Fig. 1(a). The particle is tangent to the glass.
The electric and magnetic fields of the j'th incident beam are expressed by
E0 is
Results and discussion
According to the previous section's formulation and following [37], [38], [39], [40], some numerical results are presented. The Wolfram Mathematica was used to perform numerical calculations for finding force landscapes profile and their potential energy of both Rayleigh and Mie Particle in interfering optical lattices. All-optical forces are reported in the unit . All forces are calculated in the z = a plane (considering the particle tangent to the second medium), and the x,y coordinate
Conclusion
In this paper, we studied how the number of the incident beams, their polarization, and incident angle affect force profiles landscapes for Rayleigh and Mie spherical particles in the multi-beam optical tweezers. All the simulations are performed in the Wolfram Mathematica. We investigated how the gradient, scattering, total forces, and particles' potential energy vary with the number of incident beams, incident angle, and particle size. When the number of incident beams is even, the scattering
Author statement
Our manuscript is part of the result of a research program being conducted in our group. This is an original piece of research work which is novel and is not published before and is not considered to be sent for publication elsewhere.
Declaration of Competing Interest
The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript.
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